I guess I'll contribute my lits to this thread, since video game music is quite awesome. I'll edit more in as I can think of them.
Clash on the Big Bridge - FF5 (SNES)
This is an amazing song with some extremely epic moments in it. It was so good that Nobuo Uematsu's band, the Black Mages, decided to re-do it in a rock version at one of their concerts. I love this song.
The Entire Soundtrack to Teenage Mutant Ninja Turtles: Turtles in Time (SNES)
This game is amazing, and one of the things that contributed to this is the awesome soundtrack. Honestly, listen to the music in this game and you will be impressed.
Dungeon Explorer Soundtrack (TG16)
Not many people know anything about this game, but the music in it is quite awesome. The music in the game sounded like it pushed the TG16 to its limit. I love it to death.
Cyber Core Soundtrack (TG16)
This soundtrack was another gem for the Turbo Grafx 16. Every single stage has some amazing music. If you can find the time to listen to some of the music from this game, you will not be disappointed. I'd recommend listening to stage 1 or stage 8's music, since those were definitely the best songs on the soundtrack.
Chrono Trigger Soundtrack (SNES)
This one has already been mentioned, but I feel the need to mention it again, because it's one of my favourite soundtracks of all time. I particularly like Boss Battle 2, with the extremely intense boss fights against important bosses, though I like the entire soundtrack on its own.
Katamari Damacy 1 & 2 (PS2)
I highly recommend listen to some of the songs in the soundtracks for both of these games. They're both quite cool, and the music fits extremely well with the game, and includes several different genres of music. I particularly like the "Katamari March Damacy" song (I think that's what it's called). "A Crimson Rose and a Gin Tonic", and "Cherry Blossom Color Season" are both completely awesome as well.
All Phoenix Wright/Apollo Justice Soundtracks (DS)
Yeah, these don't even need justification for being here. I like all of the intense cross-examination songs, because they add to the tension of the game so much, and it makes an element of the game that can only be brought by good music. I'm going to leave it at that.
Psychic World Soundtrack (Game Gear)
Yeah, this was a shitty portable system (though it was good for its time), but this game in particular had some great music for each of the 5 stages. The music is quite intense, and goes very well with each of the stages. :D
Grandia II Soundtrack (DC/PS2)
This game is very underrated, especially the soundtrack. There was a ton of work put into it, and it has several genres of music. I'd recommend listening to "Cancao do Povo". That song is truly awesome, and if you can understand Portuguese, it's even better since that's the language the song is in.
The Circus/Gundorada Workshop - Dark Cloud 2 (PS2)
These are two songs I particularly like from the Dark Cloud 2 soundtrack. The first song is super-happy, and I am one that likes happy music (as long as it isn't too kooky), so that's how it fits onto my list. The second song has some awesome banjo/latin guitar (I'm not sure which one). This whole game had a pretty awesome soundtrack, but I didn't play the whole game, so I don't remember all the music, since I don't listen to the soundtrack a huge amount.
Earthbound Soundtrack (SNES)
This one doesn't need justification either. The soundtrack in this game made up for the graphics by a huge amount, and contributed very much to this game, making it one of the best games of all time IMO. I very much like the song "Battle against an easy opponent" (I think that's what it's called), and I also like "High on Coffee".
That's all I can think of for now, but I'm proud of my list. :D